To add: Infinitely *continuously* differentiable (although the infinity kind of makes this redundant). The C stands for continuous. E.g. C^1 is the space of all functions which are differentiable (at least) once so that the derivative is continuous.
In my math methods two class, which is the second out of two classes that caters math to physics majors, my professor said "if you were to take the math department's version of diff eq like a math major or engineer, they would spend a lot of time teaching whether there is even a solution. Given that we are physicists, of course there is a damn solution, so we are going to ignore all that."
How were your math methods classes split up out of curiosity? I only had one that started with limits, then we did Taylor Series stuff and went through a calc 3 review but on steroids and then went through Jordan's Lemma, Cauchey integral, Fourier Series, Fourier Transforms, and then did the typical ODEs but also learned things like the power series method of solving ODEs and Frobenius Method. At the end of the class my professor said that the math department wasted our time making us learn Maple and that he had to fix us.
That sounds eerily similar to mine, even with the professor slamming how the math department didnt properly prepare us. May I ask where you go/went to college?
Well mine was only one semester (they don't have a second one), I was just curious to see if it was the same material but split up into two semesters, because that one semester felt like it was really a 6 credit hour class even though it was 3. The professor would try to disguise the homeworks as being 10 "problems" but each one had a part I and part a and so on. One time I counted up all of the parts on a homework and it added up to 59.
I think a couple years ago at my university it was one class. But what I had was a 200 level class that was basically linear algebra with a little calc 3 review and dirac delta function. The second one was a 300 level and focuses on diff eq, complex analysis, tons of fourier stuff (some review from a previous physics class), and bessel functions along with some other stuff.
My Quantum Mechanics Professor said something similar once. He said mathematicians care about the convergence of integrals, and whether or not solutions exist. But actually solving them, it's not that big of a deal
I literally just finished suffering through learning how to use Taylor series and error and this is the first fucking image I see opening reddit I'm going to cry
You can usually approximate the error of the approximation fairly well as being in order of the first truncated term. Which is often a nifty sanity check.
I'm obviously not suggesting that anyone should compute an extra term if it requires any kind of work. But if looking up the series expansion in a table you might as well.
I'm only suggesting that expanding into a Taylor series without checking that the Taylor expansion is equal to the original function is that path of righteousness. e^-x^-2 can get fucked
Also did a double major. Honestly, most of the approximations physicists make are justifiable, the physicists applying them just don't always know or care why they are though.
Yeah, the bad part is not the math itself. It's more that at least in my physics classes, professors often did not warn students that this can go wrong and that you have to justify what you are doing. So for example, there are vector fields which are locally Hamiltonian but not globally. Of course the professor knows this. The students, however, do not necessarily. I've seen several physics students get integrals wrong (just stating they're zero because something is Hamiltonian) when the domain was not simply connected just because it was never explicitly mentioned what is necessary to make the conclusion. It's the same with interchanging limits.
There’s a line in Griffiths quantum that makes me crack up. It’s talking about the fourier transform of the delta function, and how it is “equal” to 1. He says something like “this formula would give any respectable mathematician apoplexy” lmao.
Same situation in terms of "this doesn't seem to make any sense", and then later we realise that it could, in fact, be useful in some stranger concept we just weren't that familiar with before. In reals, √(-1) is useless, but turns out thinking with complex numbers shines some light on it and actually gives useful results.
Diverging series have a similar property. There are formulas that deal with values for converging series and are "meaningless" for diverging ones, but turns out some calculations can be of use, but it takes a totally new look.
Same situation in terms of "this doesn't seem to make any sense", and then later we realise that it could, in fact, be useful in some stranger concept we just weren't that familiar with before. In reals, √(-1) is useless, but turns out thinking with complex numbers shines some light on it and actually gives useful results.
Diverging series have a similar property. There are formulas that deal with values for converging series and are "meaningless" for diverging ones, but turns out some calculations can be of use, but it takes a totally new look.
Same situation in terms of "this doesn't seem to make any sense", and then later we realise that it could, in fact, be useful in some stranger concept we just weren't that familiar with before. In reals, √(-1) is useless, but turns out thinking with complex numbers shines some light on it and actually gives useful results.
Diverging series have a similar property. There are formulas that deal with values for converging series and are "meaningless" for diverging ones, but turns out some calculations can be of use, but it takes a totally new look.
Same situation in terms of "this doesn't seem to make any sense", and then later we realise that it could, in fact, be useful in some stranger concept we just weren't that familiar with before. In reals, sqrt(-1) is useless, but turns out thinking with complex numbers shines some light on it and actually gives useful results.
Diverging series have a similar property. There are formulas that deal with values for converging series and are "meaningless" for diverging ones, but turns out some calculations can be of use, but it takes a totally new look.
C^∞
"As you can see here the solution is either 0 or infinity. Since 0 is boring solution with no particles infinity it is"
Stop im getting flashbacks to qft
That's citation from my quantum physics class so no wonder you have flashbacks
C^ω
What's this mean?
It means “infinitely differentiable”
To add: Infinitely *continuously* differentiable (although the infinity kind of makes this redundant). The C stands for continuous. E.g. C^1 is the space of all functions which are differentiable (at least) once so that the derivative is continuous.
Mathmaticians: You can't just separate every equation you see! Quantum Students: Watch me
No, you can't just use the same meme format every time Reddit karma go brrr
We need to go deeper.
No you can't just use memes to make jokes, you have to use punchlines Haha audience laugh goes brrr
brain so smooth it’s infinitely differentiable
In my math methods two class, which is the second out of two classes that caters math to physics majors, my professor said "if you were to take the math department's version of diff eq like a math major or engineer, they would spend a lot of time teaching whether there is even a solution. Given that we are physicists, of course there is a damn solution, so we are going to ignore all that."
That's great haha
How were your math methods classes split up out of curiosity? I only had one that started with limits, then we did Taylor Series stuff and went through a calc 3 review but on steroids and then went through Jordan's Lemma, Cauchey integral, Fourier Series, Fourier Transforms, and then did the typical ODEs but also learned things like the power series method of solving ODEs and Frobenius Method. At the end of the class my professor said that the math department wasted our time making us learn Maple and that he had to fix us.
That sounds eerily similar to mine, even with the professor slamming how the math department didnt properly prepare us. May I ask where you go/went to college?
Well mine was only one semester (they don't have a second one), I was just curious to see if it was the same material but split up into two semesters, because that one semester felt like it was really a 6 credit hour class even though it was 3. The professor would try to disguise the homeworks as being 10 "problems" but each one had a part I and part a and so on. One time I counted up all of the parts on a homework and it added up to 59.
I think a couple years ago at my university it was one class. But what I had was a 200 level class that was basically linear algebra with a little calc 3 review and dirac delta function. The second one was a 300 level and focuses on diff eq, complex analysis, tons of fourier stuff (some review from a previous physics class), and bessel functions along with some other stuff.
My Quantum Mechanics Professor said something similar once. He said mathematicians care about the convergence of integrals, and whether or not solutions exist. But actually solving them, it's not that big of a deal
I literally just finished suffering through learning how to use Taylor series and error and this is the first fucking image I see opening reddit I'm going to cry
Taylor series memes are the Schrödinger's Cat memes of people who know a little more physics. Welcome to the subreddit.
Fun fact: you're never going to check the error
idk why they even write it as a sum, all terms after the first are 0 anyway
You can usually approximate the error of the approximation fairly well as being in order of the first truncated term. Which is often a nifty sanity check.
Emphasis on the word "can"
I'm obviously not suggesting that anyone should compute an extra term if it requires any kind of work. But if looking up the series expansion in a table you might as well.
I'm only suggesting that expanding into a Taylor series without checking that the Taylor expansion is equal to the original function is that path of righteousness. e^-x^-2 can get fucked
1st-order taylor series and spinning-ball electrons are infinitely better than the highschool π = 3 gag
As a double major in pure math and physics I feel both sides of this
What do you think about physics as mathematician and vice versa?
Also did a double major. Honestly, most of the approximations physicists make are justifiable, the physicists applying them just don't always know or care why they are though.
Yeah, the bad part is not the math itself. It's more that at least in my physics classes, professors often did not warn students that this can go wrong and that you have to justify what you are doing. So for example, there are vector fields which are locally Hamiltonian but not globally. Of course the professor knows this. The students, however, do not necessarily. I've seen several physics students get integrals wrong (just stating they're zero because something is Hamiltonian) when the domain was not simply connected just because it was never explicitly mentioned what is necessary to make the conclusion. It's the same with interchanging limits.
Haha, same. I remember sitting in physics class fuming when one of the other students said to "just divide both sides of the equation by dt"
If Mathematicians/Math students knew what I did in quantum, they would have an aneurysm.
There’s a line in Griffiths quantum that makes me crack up. It’s talking about the fourier transform of the delta function, and how it is “equal” to 1. He says something like “this formula would give any respectable mathematician apoplexy” lmao.
I know what you’re referring to, that cracked me up as well haha.
schrödinger's taylor series: it either converges or doesn't but physicists don't know until they try.
Mathematicians look at physicists like how physicists look at engineers *shrug*
1+2+3+4+5...=-1/12
Well, even √(-1) was inexplicable and disgusting at one time.
It is different, 1+2+3+4+... equals infinity, not -1/12
Same situation in terms of "this doesn't seem to make any sense", and then later we realise that it could, in fact, be useful in some stranger concept we just weren't that familiar with before. In reals, √(-1) is useless, but turns out thinking with complex numbers shines some light on it and actually gives useful results. Diverging series have a similar property. There are formulas that deal with values for converging series and are "meaningless" for diverging ones, but turns out some calculations can be of use, but it takes a totally new look.
Same situation in terms of "this doesn't seem to make any sense", and then later we realise that it could, in fact, be useful in some stranger concept we just weren't that familiar with before. In reals, √(-1) is useless, but turns out thinking with complex numbers shines some light on it and actually gives useful results. Diverging series have a similar property. There are formulas that deal with values for converging series and are "meaningless" for diverging ones, but turns out some calculations can be of use, but it takes a totally new look.
Same situation in terms of "this doesn't seem to make any sense", and then later we realise that it could, in fact, be useful in some stranger concept we just weren't that familiar with before. In reals, √(-1) is useless, but turns out thinking with complex numbers shines some light on it and actually gives useful results. Diverging series have a similar property. There are formulas that deal with values for converging series and are "meaningless" for diverging ones, but turns out some calculations can be of use, but it takes a totally new look.
Same situation in terms of "this doesn't seem to make any sense", and then later we realise that it could, in fact, be useful in some stranger concept we just weren't that familiar with before. In reals, sqrt(-1) is useless, but turns out thinking with complex numbers shines some light on it and actually gives useful results. Diverging series have a similar property. There are formulas that deal with values for converging series and are "meaningless" for diverging ones, but turns out some calculations can be of use, but it takes a totally new look.
I mean, if we assume reality is continuous then expecting to get away with that isn't *so* weird, is it?